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Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 148–157 (Mi ufa396)  

Analytic functions with smooth absolute value of boundary data

F. A. Shamoyan

Bryansk State University named after Academician Ivan Georgiyevich Petrovsky, Bezhitsckaya str. 14, 241036, Bryansk, Russia

Abstract: Let $f$ be an analytic function in the unit circle $D$ continuous up to its boundary $\Gamma$, $f(z) \neq 0$, $z \in D$. Assume that on $\Gamma$, the function $|f|$ has a modulus of continuity $\omega(|f|,\delta)$. In the paper we establish the estimate $\omega(f,\delta) \leq A\omega(|f|, \sqrt{\delta})$, where $A$ is a some non-negative number, and we prove that this estimate is sharp. Moreover, in the paper we establish a multi-dimensional analogue of the mentioned result. In the proof of the main theorem, an essential role is played by a theorem of Hardy–Littlewood type on Hölder classes of the functions analytic in the unit circle.

Keywords: analytic function, modulus of continuity, factorization, outer function.

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English version:
Ufa Mathematical Journal, 2017, 9:3, 148–157 (PDF, 349 kB); https://doi.org/10.13108/2017-9-3-148

Bibliographic databases:

UDC: 517.53
MSC: Primary 30D55, 30D15; Secondary 46E22, 47A15
Received: 10.05.2017

Citation: F. A. Shamoyan, “Analytic functions with smooth absolute value of boundary data”, Ufimsk. Mat. Zh., 9:3 (2017), 148–157; Ufa Math. J., 9:3 (2017), 148–157

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